Simultaneous equations with infinite solutions
Use this page when the working seems to collapse to the same equation and you want to understand why that means the system has infinitely many solutions.
Start here if you want the short version before reading the full method.
- A simultaneous-equations system has infinite solutions when both equations describe the same line.
- On a graph, that means every point on one line is also on the other because they are really the same line.
What this topic means and what to look for first.
This case can feel confusing because the algebra does not end with one pair of values. Instead, it shows that the two equations are dependent on each other.
The graph interpretation makes it clearer: if both equations are the same line, there is not one shared point but infinitely many.
One reliable route through the topic.
- 1Solve the system as usual by elimination or substitution.
- 2Watch for the equations collapsing into an identity such as 0 = 0.
- 3Interpret that identity as meaning the two equations describe the same line rather than a single point.
- 4Explain the answer as infinitely many solutions instead of forcing one x and y pair.
See the method in action.
x + y = 5 and 2x + 2y = 10
- The second equation is just the first one multiplied by 2.
- Elimination collapses to 0 = 0, which is an identity rather than a contradiction.
- That means the system has infinitely many solutions.
y = 3x - 1 and 2y = 6x - 2
- Rearrange the second equation by dividing by 2 to get y = 3x - 1.
- Now both equations are exactly the same line.
- So every point on that line is a solution, giving infinitely many solutions.
Things that commonly send the method off track.
- Thinking 0 = 0 means you have made an error, when it may actually describe the system correctly.
- Trying to force one pair of values even though the two equations are the same line.
- Missing that one equation is only a scaled version of the other.
Use a short verification pass before moving on.
- Check whether one equation is just a multiple or rearrangement of the other.
- If the system reduces to 0 = 0 consistently, interpret it as infinitely many shared solutions rather than a failed solve.
Try a few variations before switching to a calculator or solver tool.
- x + y = 6 and 2x + 2y = 12
- y = 4x + 3 and 2y = 8x + 6
- 3x - y = 7 and 6x - 2y = 14
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Short answers worth checking.
It means the two equations describe the same line, so every point on that line satisfies both equations.
A result like 0 = 0 after elimination is a strong sign that the equations are dependent and represent the same line.
No. In simultaneous equations, it can be the correct sign that the system has infinitely many solutions.
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